Nearly two dozen Republicans who have publicly questioned or disputed the 2020 election results are running for secretary of state across the country, in some cases with direct encouragement from former President Donald J. Trump’s allies.

Their candidacies have alarmed watchdog groups, Democrats, and some fellow Republicans, who are concerned that if elected to positions that exist primarily to safeguard and administer the democratic process, these Trump supporters will use those positions to undermine it — whether by outright subverting an election or sowing doubts about any local, state, or federal elections their party loses.

Secretaries of state worked in relative anonymity for decades, setting regulations and enforcing rules governing how elections were administered by local counties and boards. Some held their jobs for many years and viewed themselves not as politicians but as bureaucrats in chief, tending to such arcane responsibilities as keeping the state seal or maintaining custody of state archives.

All of that changed in the aftermath of the 2020 presidential election.

Mr. Trump and his allies put pressure on Republican secretaries of state, election board members, and other officials in battleground states to overturn President Biden’s defeat in the two months between Election Day and Congress’ certification of his victory. In a phone call that is now the subject of an Atlanta grand jury investigation into Mr. Trump’s actions in Georgia, the former president instructed Georgia Secretary of State Brad Raffensperger to “find 11,780 votes” — the margin by which Mr. Trump lost the state to Mr. Biden.

That intense focus on a once-obscure state-level office has dramatically altered its place in American politics — as well as the pool of candidates it attracts. This year’s campaigns for secretary of state are attracting more money, attention, and outspokenly partisan candidates than ever before.

According to States United Action, a nonpartisan group that tracks secretary of state races across the country, 21 candidates opposing Mr. Biden’s victory are running for secretary of state in 18 states.

Many of the election deniers are running in solidly red states, where their actions are less likely to sway a presidential election. However, several others have formed the America First slate and are running in states won by Mr. Biden in 2020, including the crucial battleground states of Michigan, Arizona, and Nevada.

Members of the coalition are coordinating talking points, staff members, and fund-raising efforts — an unusual level of collaboration for down-ballot candidates from different states. They have a good chance of winning the Republican primaries in those battleground states, as well as in slightly bluer Colorado and heavily Democratic California.

If they win the primaries, their chances in November may be heavily reliant on how well Republicans perform in the midterm elections, given voters’ proclivity to vote for down-ballot candidates from the same party as their choices for governor or senator. While local election officials typically oversee the counting of individual ballots and state legislatures approve Electoral College slates, secretaries of state frequently certify elections and set the tone for how elections are conducted. Their responsibilities in election management typically include distributing voter registration cards, allocating voting machines, educating voters, auditing election results, and ordering recounts.

If secretaries of state had followed Mr. Trump’s lead in the last election, they could have tipped the scales of fair elections by forcing polling places to close, removing ballot drop boxes, or withholding other resources that could have made voting easier in heavily Democratic precincts. Worse, critics argue, they could have questioned, or even refused to certify, Mr. Biden’s victories.